Complex numbers

A complex number is a number of the form z = a + bi, with a, b ∈ ℝ and i^{2} = −1. As a set, the complex numbers are written ℂ. Addition and multiplication are defined by

(a + bi) + (c + di) = (a + c) + (b + d)i
(a + bi)(c + di) = (ac − bd) + (ad + bc)i.

(These follow the usual algebraic rules for square roots.) A complex number is represented geometrically by an element of ℝ^{2}: a + bi ↦ \binom{a}{b}. a = Re(z), b = Im(z).

Let’s investigate the geometric interpretations of adding and multiplying in ℂ. • Addition is just adding vectors in ℝ^{2}.

• Multiplication rotates and/or scales.

Examples. Multiply by i. Multiply by 2i. Multiply by 1 + i. Multiply by \sqrt{2}/2 + i\sqrt{2}/2. Multiply by 1/2 − i\sqrt{3}/2. Multiply by −1.

Note that both rotating and scaling are linear transformations of ℝ^{2}.

We know that a point of ℝ^{2} can be represented by polar coordinates:

\binom{x}{y} ↔ \binom{r cos θ}{r sin θ},

where r = \sqrt{x^{2} + y^{2}} and θ is the angle (measured counterclockwise) between \binom{1}{0} and \binom{x}{y}. Applying this to complex numbers, we have the polar form:

z = r(cos θ + i sin θ).

r is called the modulus (or absolute value) of z, and is written r = |z|. θ is called an argument of z. Of course, θ could have many values, so we usually choose to have it in one of the intervals [0, 2π) or (−π, π].

So we know we can think of complex numbers as real vectors with 2 coordinates. But we’ve also seen that multiplying by a complex number produces a linear transformation. What are the matrices of these maps?

z ↦ iz  \binom{x}{y} ↦ (\begin{matrix}0 & −1 \\ 1 & 0 \end{matrix}) \binom{x}{y}

z ↦ (\frac{\sqrt{2}}{2} + i\frac{\sqrt{2}}{2}) z  \binom{x}{y} ↦ (\begin{matrix}\sqrt{2}/2 & −\sqrt{2}/2 \\ \sqrt{2}/2 & \sqrt{2}/2 \end{matrix}) \binom{x}{y}

A matrix of the form (\begin{matrix}cos θ & − sin θ \\ sin θ & cos θ \end{matrix}) is called a rotation matrix.

A matrix of the form (\begin{matrix}r & 0 \\ 0 & r \end{matrix}) is called a scalar matrix. Scalar matrices commute with all

other matrices.

So we can also think of complex numbers as matrices, the product of a scalar matrix and a rotation matrix (in either order). This preserves the multiplicative structure. Check for yourself that this way of thinking also preserves the additive structure. Proposition. Let a be a complex number. Then the map z ↦ az, thought of as a map ℝ^{2} → ℝ^{2}, is linear, and is the composition of scaling and rotation.